A. Did I tell you that I was going to Hawaii next month? and
B, Did I tell you that I am going to Hawaii next month? but less so with

C. Did I tell you that I will go to Hawaii next month? This last one is possible but the register is different (more formal) and makes the trip sound more like a royal visit than a holiday.

Change next month to the following month and you don't alter the rules about tenses in reported speech, though you do alter the perceived time status of the sentence.

For me all three a distinctly possible: A. is maybe the most likely. B. is fine if your plans to go are still intact. C. makes you sound as though you are a head of state, unless you are using will with its force of to be willing.

There are a lot of issues here. There are circumstances, as Velisarius points out, where B would be used rather than A.

Did I tell you that I ______ to Hawaii next month?
A. was going
B. am going
I don't understand why the answer is A.

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The grammar book is correct. The that-clause represents the thought in the person's mind at the time of speaking and therefore is governed by the time-context of the original statement.

This is the traditional grammatical analysis which I maintain despite the argument of some people that the content of the that-clause should be considered outside its time-frame.

On the traditional view, the following correct options are available:

1(a) Have I told you that I am going to Hawaii next month? (Main verb: present perfect. Time-context: present)
(b) Did I tell you that I was going to Hawaii next month? (Main verb: past simple. Time-context: past)

2(a) Have I told you that I will go to Hawaii next month? (Main verb: present perfect. Time-context: present)
(b) Did I tell you that I would go to Hawaii next month? (Main verb: past simple. Time-context: past)

Note:
1(a) and 2(a) imply that the plan or intention of the speaker still holds good.
In 1(b) and 2(b), it is not implied that the plan or intention still holds good, but it is not excluded either.
The (b) sentences allow both possibilities: (1) that the plan or intention is unchanged and (2) that the plan or intention is been changed.